Gradients in the curvature of space-time

This is my first post. I'm a newbie to general relativity, but I think I'm getting the hang of it thanks to some helpful professors at UC Berkeley.

From what I understand, and now fully believe, there are no external forces applied to an object that is free falling in space-time -- regardless of the degree of curvature in that space-time. An object can transition from relatively flat space-time (i.e. far away from other massive objects) to curvier space-time (e.g. getting close to a planet) and there is no external force. Therefore, this object cannot use simple force meters (one's own skin, or a spring, or similar) to detect whether or not he/she is about to smash into an, for example, atmosphere-free planet (without looking).

However, it does seem that there is an internal force that tends to tear an object apart due to gradients in the curvature of space-time. I believe that this is what tore apart the Shoemaker-Levy 9 comet before it smashed into Jupiter. The easiest way for me to visualize this is to imagine a string of very small beebees heading toward a planet. The beebees that are closest to the planet are subjected to a larger amount of space-time curvature, so an increase in the distance between the beebees occurs with the closest-to-the-planet ones getting farther apart than the ones behind them. Now, instead of beebees it is a person (feet toward the planet), then the toes will tend to be pulled away from the head.

So, it seems that this gradient in curvature is the only method that one could use, when free falling, to detect the magnitude of curvature in one's local region space-time. Does this make sense?

Also, has anyone ever "constructed" a mass (e.g. an oddly-shaped planet), either mathematically or via simulation, that creates a region of space with a constant curvature of space-time?

mmm.. you are saying using "tidal forces" (over a period of time of falling) to detect the magnitude of curvature in one's local region of spacetime. yeah, make sense to me (although me no expert in GR). Not sure whether you can only get the radial gradient from that

Attached Files:

If you're up to the maths, the texts most people use are Gravitation by Misner, Thorne and Wheeler, or General Relativity by Wald. The first, IMO, is brilliant, eccentric, and at times funny (I guess I am that sad). It also goes to great pains to try and impart intuitive understanding of GR and curvature in general. So if you want some pictures -- go for MTW!

As far as I understand, the only three types of geometry with constant curvature are elliptical, Euclidean and hyperbolic geometries. So starting with the metric tensor for each of these, if you went through and worked out the respective Einstein tensors, then by the EFE's find the energy-momentum tensor required for such a geometry.